A Loewner Equation for Infinitely Many Slits
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Abstract
It is wellknown that the growth of a slit in the upper halfplane can be encoded via the chordal Loewner equation, which is a differential equation for schlicht functions with a certain normalisation. We prove that a multiple slit Loewner equation can be used to encode the growth of the union \(\Gamma \) of multiple slits in the upper halfplane if the slits have pairwise disjoint closures. Under certain assumptions on the geometry of \(\Gamma \), our approach allows us to derive a Loewner equation for infinitely many slits as well.
Keywords
Loewner theory Chordal Loewner equation Slit domain Infinitely many slitsMathematics Subject Classification
30C20 30C551 Introduction and Main Result
Definition 1.1
A bounded set \(A\subseteq \mathbb {H}\) is called a (compact) \(\mathbb {H}\) hull (or for short: hull) if \({{\mathrm{clos}}}(A)\cap \mathbb {H}=A\) and \(\mathbb {H}{\setminus } A\) are simply connected. A hull A is called a slit if there is a homoemorphism \(\gamma :[0,1]\rightarrow {{\mathrm{clos}}}(A)\), such that \(\gamma (0)\in \mathbb {R}\) and \(\gamma (0,1]\subseteq A\), where \(\gamma \left( 0,t\right] \) denotes the image of the halfopen interval \(\left( 0,t\right] \) under \(\gamma \). In this case, we say \(\gamma \) parametrises A. A multislit is a possibly finite sequence of slits \(\Gamma _{j}\), such that \(\bigcup _{j}\Gamma _{j}\) is a hull. Given \((\Gamma _{j})_{j}\), we let \(\Gamma : =\bigcup _{j}\Gamma _{j}\), and also call \(\Gamma \) a multislit. Moreover, if for a multislit \(\Gamma \), the set \({{\mathrm{clos}}}\Gamma _{j}\) can be separated from \({{\mathrm{clos}}}(\Gamma {\setminus }\Gamma _{j})\) by open sets for each j, then \(\Gamma \) is called admissible (see Fig. 1). If we wish to emphasise that a multislit consists of only finitely many slits, then we speak of an n slit. In what follows, every multislit \(\Gamma \) is assumed to be admissible.
Recently, several authors, in particular mathematical physicists gazing towards conformal field theory, have studied a Loewner equation for multiple slits, to generate growing hulls or n slits; see, for example, [5, 8, 15, 16]. However, the following geometric question has apparently received little attention: for what kind of parametrisations can any given multislit be encoded in a Loewner equation? In the radial case, there are some results for finitely many slits, see [2, 3]. In the chordal case, it is, to the best of the authors’ knowledge, only known that for n slits, there exists a certain (not effectively computable) parametrisation, such that a generalised Loewner equation is satisfied, see [14, Theorem 1.1]. To state our results, we recall the following wellknown fact (cf. [10, p. 69]).
Proposition 1.2
For each hull A, there is a unique biholomorphism \(g_{A}:\mathbb {H}{\setminus } A\rightarrow \mathbb {H}\) satisfying (1.1). Moreover, \(\lim _{z\rightarrow \infty }z(g_{A}(z)z)\) exists, and is called the halfplane capacity \({{\mathrm{hcap}}}(A)\) of A.
Furthermore, we need the following notation.
Definition 1.3
Let \(\Gamma \) be a multislit. We call \(\gamma =(\gamma _{j})_{j}\) a parametrisation of \(\Gamma \) if \(\gamma _{j}\) is a parametrisation of \(\Gamma _{j}\) for every j. By a slight abuse of notation, we let \(\Gamma _{t}: =\bigcup _{j}\gamma _{j}\left( 0,t\right] \).^{1} We call a parametrisation \(\gamma \) of \(\Gamma \) a Loewner parametrisation of \(\Gamma \) if \(t\mapsto {{\mathrm{hcap}}}(\Gamma _{t})\) is Lipschitz continuous for \(t\in \left[ 0,1\right] \).
Corollary 3.3 will show that these “normalised” parametrisations can be achieved to encode a given multislit \(\Gamma \) in a Loewner equation. Given a multislit \(\Gamma \), we write \(g_{t}: =g_{\Gamma _{t}}\), and denote by \(h_{t}\) the inverse of \(g_{t}\). Consequently, we also denote by \(h_{\Gamma }\) the inverse of \(g_{\Gamma }\). Our main result is the following:
Theorem 1.4
 (1)
\(0\le \lambda _{j}(t)\le L\) almost everywhere,
 (2)
Each \(\lambda _{j}\) is measurable,
 (3)
\( \sum _{j}\lambda _{j}(t) = \frac{\partial }{\partial t}{{\mathrm{hcap}}}\Gamma _{t} \) almost everywhere,
Informally speaking, the weight function \(\lambda _{j}(t)\) corresponds to the “speed” in which \(\gamma _{j}(t)\) grows at the time \(t\in [0,1]\), and the driving function \(U_{j}=g_t\circ \gamma _j\) keeps track of the position of the tip \(\gamma _{j}(t)\) of the slit \(\gamma _{j}(0,t]\).
Moreover, we would like to mention that the a.e. differentiability part of the theorem above is a consequence of a more general phenomenon occurring in Loewner theory, cf. [7, Theorem 3], [4, Theorem 1.1]. Furthermore, in the 1slit case, one recovers from Theorem 1.4 the ordinary Loewner equation, which relates the existence of the derivative \(\dot{g}_{t}\) to the existence of the weight functions \(\lambda _{j}\).
Remark 1.5
Let us mention that the nslit case in Theorems 1.4 and 5.2 has been proved in [1, Theorem 2.36, 2.54] (see also [2, 3] for results in the radial case). Moreover, Theorem 1.4 is best possible in the sense that \(t\mapsto g_{t}(z)\) (\(z\in \mathbb {H}{\setminus }\Gamma \)) is in general not (everywhere) differentiable, see Remark 5.3.
The paper is structured as follows. First, we collect some basic tools in Sect. 2. These are needed to study the difference quotient of \(t\mapsto g_{t}(z)\), where \(g_{t}\) is the map from Theorem 1.4. To this end, we use classical results from (geometric) function theory, e.g., the theory of prime ends, kernel convergence, normal families, and the Nevanlinna representation. In Sect. 3, we construct the driving functions. In Sect. 4, we construct functions that will later on turn out to be the weight functions. These are the main problems when passing from the 1slit Loewner equation to the multislit version. For overcoming this obstacle, we use tools from Lipschitz analysis. Eventually, we put the pieces together in Sect. 5 to derive the results stated above.
2 Preliminaries and basic tools
Let us mention, for the sake of clarity, that we equip \(\hat{\mathbb {C}}\) with its natural topology; in particular, the boundary of \(\mathbb {H}\) is understood to contain the point \(\infty \). Moreover, we fix an arbitrary admissible multislit \(\Gamma \) with Loewner parametrisation \(\gamma \) throughout Sects. 2–5.
The next theorem deals with biholomorphic extensions of the maps \(g_{\Gamma }\) and \(h_{\Gamma }\), and is a direct consequence of the wellknown Schwarz reflection principle combined with the classical theory of prime ends (cf. [12]).
Theorem 2.1
Let \(\Gamma \) be an admissible multislit. Then, any given map \(g_{\Gamma }\) extends to a biholomorphism from \(\hat{\mathbb {C}}{\setminus }({{\mathrm{clos}}}(\Gamma )\cup \Gamma ^{*})\) onto \(\hat{\mathbb {C}}{\setminus }\mathcal {C}\) for some \(\mathcal {C}\subseteq \mathbb {R}\), where \(\Gamma ^{*}\) denotes the complex conjugate of the set \(\Gamma \).
Moreover, we need the following fact about the relationship between the size of the preimage of a boundary point and the topology of the boundary of the image domain of a biholomorphism (see [12, Chapter 2, Proposition 2.5]).
Theorem 2.2
Let \(h:\mathbb {H}\rightarrow D\) be a biholomorphism, where D has locally connected boundary \(\partial D\). Fix \(w_{0}\in \partial D\), and consider the preimage \( \mathcal {W} : =h^{1}(\{w_{0}\})\subseteq \hat{\mathbb {R}} : =\mathbb {R}\cup \{\infty \} \) of \(w_{0}\). Then, there is a bijection \(\mathcal {C}\mapsto h\left( \mathcal {C}\right) \) between the connected components \(\mathcal {C}\) of \(\hat{\mathbb {R}}{\setminus }\mathcal {W}\) and the connected components of \(\partial D{\setminus }\{ w_{0}\}\). In particular, \(\mathcal {W}\) consists of precisely n (pairwise distinct) points if \(\partial D{\setminus }\{ w_{0}\}\) has n connected components.
Definition 2.3

\(\mathcal {J}_{t,T,j} : =g_{t}(\gamma _{j}\left( t,T\right] )\subseteq \mathbb {H}\),

\(\mathcal {\overline{J}}_{t,T,j} : =g_{t}(\gamma _{j}[t,T]) \) (\(g_{t}(\gamma _{j}(t))=:U_{j}(t)\) is welldefined by Theorem 2.2),

\(\mathcal {J}_{t,T} : =\bigcup _{j}\mathcal {J}_{t,T,j} \),

\(\mathcal {\overline{J}}_{t,T} : =\bigcup _{j}\overline{\mathcal {J}}_{t,T,j} \),

\(\mathcal {C}_{t,T} : =\bigcup _{j}\mathcal {C}_{t,T,j}\subseteq \mathbb {R}\) where \(\mathcal {C}_{t,T,j}\) is the preimage of \(\gamma _{j}\left[ t,T\right] \) under \(h_{T}\) (in the sense of Theorem 2.2, and observe that the normalisation in Proposition 1.2 implies that the point \(\infty \) is not contained in \(\mathcal {C}_{t,T,j}\)).
We can deduce the following properties for these quantities.
Lemma 2.4
 (1)
the function \(\varphi _{t,T}:\mathbb {H}\rightarrow \mathbb {H}{\setminus }\mathcal {J}_{t,T}\) admits a continuous extension to the boundary,
 (2)
the sets \(\mathcal {C}_{t,T,k}\) are pairwise disjoint intervals.
Proof
We first prove (1). Recall that a biholomorphism from \(\mathbb {H}\) onto a domain D admits a continuous extension if and only if \(\partial D\) is locally connected (see [12, Theorem 2.1]). To prove (1), it suffices to show that \(\mathcal {J}_{t,T}\) is locally pathconnected as \( \partial \left( \mathbb {H}{\setminus }\mathcal {J}_{t,T}\right) =\hat{\mathbb {R}}\cup \mathcal {J}_{t,T} \). Therefore, we only need to show that any given \(\mathcal {J}_{t,T,k}\) can be separated from \({{\mathrm{clos}}}(\mathcal {J}_{t,T}{\setminus }\mathcal {J}_{t,T,k})\) by some neighbourhood U. However, this is evident, since \(\Gamma \) was assumed to be an admissible multislit.
We now show (2). By (1), the map \(\varphi _{t,T}\) extends continuously to \(\hat{\mathbb {R}}\). Note that by the pathconnectedness of \(\mathbb {H}{\setminus }\mathcal {J}_{t,T}\), we can consider a simple curve \(J_{k}^{}\) which connects the tip of a given \(\mathcal {J}_{t,T,k}\), i.e., the point \(g_{t}(\gamma _{k}(T))\), with its starting point, i.e., the point \(g_{t}(\gamma _{k}(t))\), from the left.^{3} The preimage \(\tilde{J}_{k}^{}: =\varphi _{t,T}^{1}(J_{k}^{})\) is a simple curve in \(\mathbb {H}\) that connects two distinct boundary points \(\alpha ,\beta \in \mathbb {R}\). Denote by \(\omega _{k}^{},\Omega _{k}^{}\) the interior of \(\left[ \alpha ,\beta \right] \cup \tilde{J}_{k}^{}\), and of \(\mathcal {J}_{t,T,k}\cup J_{k}^{}\), respectively. We can extend the homoemorphism \(\varphi _{t,T}_{\omega _{k}^{}}:\omega _{k}^{}\rightarrow \Omega _{k}^{}\) to a homoemorphism from \({{\mathrm{clos}}}\omega _{k}^{}\) onto \({{\mathrm{clos}}}\Omega _{k}^{}\). Then, the preimage of \(\mathcal {J}_{t,T,k}\) under \(\varphi _{t,T}_{{{\mathrm{clos}}}\omega _{k}^{}}\) has to be the interval \(\left[ \alpha ,\beta \right] \). By applying the same reasoning to a curve \(J_{k}^{+}\) that connects the tip of \(\mathcal {J}_{t,T,k}\) with its starting point from the right, we get that the preimage of \(\mathcal {J}_{t,T,k}\) under \(\varphi _{t,T}\mid _{{{\mathrm{clos}}}\omega _{k}^{+}}\) is an interval of the form \(\left[ \beta ,\alpha '\right] \). In view of Theorem 2.2, we find that \( \mathcal {C}_{t,T,j} =\left[ \alpha ,\beta \right] \cup \left[ \beta ,\alpha '\right] \), so \(\mathcal {C}_{t,T,j}\) is an interval. Theorem 2.2 yields that these intervals are disjoint.\(\square \)
Furthermore, there is a simple but crucial integral representation for \(\varphi _{t,T}\):
Lemma 2.5
 (1)
it holds that \(\varphi _{t,T}\circ \varphi _{T,s}=\varphi _{t,s}\),
 (2)for all \(z\in \mathbb {H}\):$$\begin{aligned} \varphi _{t,T}(z) = z+\frac{1}{\pi }\int _{\mathcal {C}_{t,T}} \frac{{\text {Im}}(\varphi _{t,T}(\xi ))}{\xi z} \,\mathrm {d}\xi = z+\frac{1}{\pi }\sum _{k}\int _{\mathcal {C}_{t,T,k}} \frac{{\text {Im}}(\varphi _{t,T}(\xi ))}{\xi z}\,\mathrm {d}\xi , \end{aligned}$$
 (3)we have that$$\begin{aligned} {{\mathrm{hcap}}}(\Gamma _{T}){{\mathrm{hcap}}}(\Gamma _{t}) =\frac{1}{\pi }\int _{\mathcal {C}_{t,T}} {\text {Im}}(\varphi _{t,T}(\xi ))\,\mathrm {d}\xi . \end{aligned}$$
Proof
(1) is a simple calculation.
Now, we are in the position to conclude a simple, but very useful lemma.
Lemma 2.6
Proof
Using the previous lemma, we can derive a crucial fact about the differentiability of \(\tau \mapsto g_{\tau }(z)\). Namely, we have:
Corollary 2.7
Proof
3 Driving Functions
Lemma 3.1
Proof
If \(0\in {{\mathrm{clos}}}(A)\), then the claim follows from [10, Corollary 3.44]. By taking \(c\in \mathbb {R}\), such that \(B: =Ac \) satisfies \(0\in {{\mathrm{clos}}}(B)\), we can deduce the general case from \(g_{B}(z)=g_{A}(z+c)c\) and \(g_{A}(z+c)(z+c)=g_{B}(z)z\).\(\square \)
Theorem 3.2
If T respectively t is fixed, then, for fixed k, there is a \(\delta >0\), such that for all \(t\in \left[ T\delta ,T\right] \), respectively, \(T\in \left[ t,t+\delta \right] \), we can separate \(\mathcal {C}_{t,T,k}\) from \(\mathcal {C}_{t,T}{\setminus }\mathcal {C}_{t,T,k}\) by a (fixed) open set for any k.
Proof
We consider the case \(t\nearrow T\). Since \(\mathcal {C}_{t,T,k}=h_{T}^{1}(\gamma _{k}[t,T])\) is becoming smaller as \(t\nearrow T\), it suffices to separate \(\mathcal {C}_{t,T,k}\) from \(\mathcal {C}_{t,T}{\setminus }\mathcal {C}_{t,T,k}\) for some t. Note that we can separate \(\mathcal {J}_{t,T,k}\) from \(\mathcal {J}_{t,T}{\setminus }\mathcal {J}_{t,T,k}\). Hence, by continuity of \(\varphi _{t,T}\), the assertion is clear in the case of \(t\nearrow T\). In the remaining case, we can separate \(\mathcal {J}_{t,T,k}=g_{t}(\gamma _{k}(t,T])\), which are getting smaller from \(\mathcal {J}_{t,T}{\setminus }\mathcal {J}_{t,T,k}\) by a simple curve J. Using Carathéodory’s Kernel theorem,^{6} we get that the simple curves \(\tilde{J}_{t,T,k}: =\varphi _{t,T}^{1}\circ J\) converge to J, which separates \(\mathcal {C}_{t,T,k}\) from \(\mathcal {C}_{t,T}{\setminus }\mathcal {C}_{t,T,k}\).\(\square \)
The next corollary demonstrates how one can normalise a given parametrisation of a multislit.
Corollary 3.3
Let \(\Gamma \) be an admissible multislit with parametrisation \(\gamma =(\gamma _{j})\), such that \(t\mapsto {{\mathrm{hcap}}}\Gamma _{t}\) is strictly increasing. Then, there exists a Loewner parametrisation of \(\Gamma \).
Proof
Now, we can characterise the limit behaviour of \(\mathcal {C}_{t,T,k}\) and \(\mathcal {J}_{t,T,k}\) as \(t\nearrow T\), or \(T\searrow t\) as follows:
Lemma 3.4
 (1)
\(\mathcal {C}_{t,T,k}\) shrinks^{7} to \(U_{k}\left( T\right) \) as \(t\nearrow T\).
 (2)
\(\mathcal {C}_{t,T,k}\) tends to \(U_{k}(t)\) as \(T\searrow t\).
 (3)
\(\mathcal {\overline{J}}_{t,T,k}\) shrinks to \(U_{k}(t)\) as \(T\searrow t\).
 (4)
\(\mathcal {J}_{t,T,k}\) tends to \(U_{k}\left( T\right) \) as \(t\nearrow T\).
To prove this, we recall the following lemma (cf. [6, Lemma 2.1]):
Lemma 3.5
 (1)
\(K_{1}\subseteq B_{2{{\mathrm{diam}}}(K_{2})}(w_{0})\) for every \(w_{0}\in K_{2}\),
 (2)
\(K_{2}\subseteq B_{2{{\mathrm{diam}}}(K_{1})}(z_{0})\) for every \(z_{0}\in K_{1}\).
Proof of Lemma 3.4
(1) and (3): By definition, we have \(\mathcal {C}_{t,T,k}=h_{T}^{1}(\gamma _{k}\left[ t,T\right] )\) and \(\mathcal {\overline{J}}_{t,T,k}=g_{t}(\gamma _{k}\left[ t,T\right] )\), so the assertions are clear.
To conclude that \(\mathcal {J}_{t,T,k}\) tends to \(U_{k}\left( T\right) \) as \(t{\nearrow } T\), it is enough to show that for any given sequence \(t_{n}{\nearrow } T\), there is a subsequence for which this claim holds. After choosing an appropriate subsequence which we denote again by \((t_{n})_{n}\), the maps \(g_{\mathfrak {J}_{k,t_{n},T}}\) tend to some schlicht g on a compact set K containing \(\mathcal {J}_{t,T,k}\). Therefore, \(\mathcal {\tilde{J}}_{t_{n},T,k}=g_{\mathfrak {J}_{k,t_{n},T}}(\mathcal {J}_{t_{n},T,k})\) implies by taking \(n\rightarrow \infty \) that \(\mathcal {J}_{t_{n},T,k}\) has to converge to some point. By arguing similarly, for \(k'\ne k\), we get that for some appropriate subsequence of \((t_{n})_{n}\), all \(\mathcal {J}_{t_{n},T,k'}\) converge to points. Hence, \(g_{\mathfrak {J}_{k,t_{n},T}}\) converges to \(\mathrm {id}_{\mathbb {C}}\). This implies that \(\mathcal {J}_{t_{n},T,k}\) tends to \(U_{k}(T)\).\(\square \)
The result above immediately implies the following important corollary.
Corollary 3.6
Let \(\Gamma \) be an admissible multislit with Loewner parametrisation \(\gamma \). Then, the driving functions \(U_j:[0,1]\rightarrow \mathbb {R}\), given by Definition 2.3, are continuous.
Proof
For \(0\le t\le T\le 1\), we get \(U_{j}(t)\in \mathcal {C}_{t,T,j}\) and \(U_{j}\left( T\right) \in \overline{\mathcal {J}}_{t,T,j}\). Therefore, Lemma 3.4 implies the right continuity and left continuity of \(t\mapsto U_{j}(t)\), and guarantees, furthermore, that \(\lim _{t\nearrow \tau }U_{j}(t)=U_{j}(\tau )=\lim _{T\searrow \tau }U_{j}(T)\).\(\square \)
4 Weight Functions
Now, we are in the position to define the weight functions from Theorem 1.4. In view of (1.2), we define these functions, roughly speaking, as the “residues” of the derivative \(z\mapsto \dot{\varphi }_{t,1}(z)\) in the “dynamical boundary points” \(U_{k}(t)\). However, doing so requires some involved analysis.
Theorem 4.1
Proof
 (i)Let \(0\le \tau \le t\). Using Lemma 2.6 with the abbreviationwe find that$$\begin{aligned} \xi _{j,k}(\tau ): ={\text {Re}}\frac{1}{\varphi _{t,1}(u_{k})\xi _{j}(\tau )}+i{\text {Im}}\frac{1}{\varphi _{t,1}(u_{k})\xi _{j}'(\tau )},\quad \xi _{j}(\tau ),\xi _{j}'(\tau )\in \mathcal {C}_{\tau ,t,j}, \end{aligned}$$Since \({\text {Im}}\xi _{j,k}(\tau )\ne 0\) is strictly negative, we can define the quantity$$\begin{aligned} D_{k}(t,\tau ): =\frac{\varphi _{t,1}(u_{k})\varphi _{\tau ,1}(u_{k})}{t\tau }&=\sum _{j}\xi _{j,k}(\tau )\frac{1}{t\tau }\frac{1}{\pi }\int _{\mathcal {C}_{\tau ,t,j}}{\text {Im}}(\varphi _{\tau ,t}(\xi ))\,\mathrm {d}\xi . \end{aligned}$$(4.3)Now let \(w_{k,j}(\tau ): =\xi _{j,k}(\tau )\xi _{1,k}(\tau )^{1}\), and note that$$\begin{aligned} \Delta _{k}(\tau ): =\frac{\varphi _{t,1}(u_{k})\varphi _{\tau ,1}(u_{k})}{(t\tau )\xi _{1,k}(\tau )},\quad \tau \in \left[ 0,t\right) . \end{aligned}$$(4.4)Since, by Lemma 3.4, \(\xi _{j}(\tau )\) and \(\xi _{j}'(\tau )\) both converge to \(U_{j}(t)\) as \(\tau {\nearrow } t\), Corollary 2.7 provides the existence of$$\begin{aligned} \Delta _{k}(\tau )=\frac{1}{t\tau }\frac{1}{\pi }\int _{\mathcal {C}_{\tau ,t,1}}{\text {Im}}\varphi _{\tau ,t}(\xi )\,\mathrm {d}\xi +\sum _{j\ne 1}w_{k,j}(\tau )\frac{1}{t\tau }\frac{1}{\pi }\int _{\mathcal {C}_{\tau ,t,j}}{\text {Im}}\varphi _{\tau ,t}(\xi )\,\mathrm {d}\xi . \end{aligned}$$(4.5)$$\begin{aligned} \lim _{\tau \nearrow t}w_{k,j}(\tau )=\frac{\varphi _{t,1}(u_{k})U_{1}(t)}{\varphi _{t,1}(u_{k})U_{j}(t)}. \end{aligned}$$(4.6)
 (ii)We claim that for any \(\varepsilon >0\), there is some \(\delta >0\), and \(n_{0}\in \mathbb {N}\), such that for all \(j\ne 1\) and \(\tau \in \left[ t\delta ,t\right] \), we have \(w_{k,j}(\tau )<\varepsilon \) for \(k\ge n_{0}\). To see this, let \( K: =\min \{\mathrm {dist}(\mathcal {C}_{\tau ,t,j},\mathcal {C}_{\tau ,t,1})\,:\,j\ne 1\}, \) and note that we have \(K>0\), by Theorem 3.2. Therefore, we use the estimateand$$\begin{aligned} \xi _{j,k}(\tau )&\le 2\Bigl (\min _{\xi \in \mathcal {C}_{\tau ,t,j}}\xi \varphi _{\tau ,1}(u_{k})\Bigr )^{1}, \end{aligned}$$Hence, for k large enough, \(\tau \) sufficiently close to t and \(j\ne 1\), we conclude that$$\begin{aligned} \min _{\xi \in \mathcal {C}_{\tau ,t,j}}\xi \varphi _{\tau ,1}(u_{k})\ge \min _{\xi '\in \mathcal {C}_{\tau ,t,j},\xi \in \mathcal {C}_{\tau ,t,1}}\xi \xi '\max _{\xi '\in \mathcal {C}_{\tau ,t,1}}\xi '\varphi _{\tau ,1}(u_{k}). \end{aligned}$$implying that \(\xi _{j,k}(\tau )\le \frac{4}{K}\) and consequently$$\begin{aligned} \min _{\xi \in \mathcal {C}_{\tau ,t,j}}\xi \varphi _{\tau ,1}(u_{k})\ge \frac{K}{2}, \end{aligned}$$which shows that \(w_{k,j}(\tau )<\varepsilon \).$$\begin{aligned} w_{k,j}(\tau )=\frac{\xi _{j,k}(\tau )}{\xi _{1,k}(\tau )}\le \frac{4}{K\xi _{1,k}(\tau )}, \end{aligned}$$
 (iii)Next, we claim that the sequence defined byis a Cauchy sequence. Let L denote the Lipschitz constant of \(t\mapsto {{\mathrm{hcap}}}(\Gamma _{t})\). By employing (4.5), Lemma 2.5 (3), and Step (ii), we deduce that$$\begin{aligned} a_{k}: =\lim _{\tau \nearrow t}\Delta _{k}(\tau )=\dot{\varphi }_{\tau ,1}(u_{k})(\varphi _{t,1}(u_{k})U_{j}(t)) \end{aligned}$$(4.7)thereby proving the claim.$$\begin{aligned} a_{M}a_{m}=&\biggl \lim _{\tau \nearrow t}\sum _{j\ne 1}(w_{M,j}(\tau )w_{m,j}(\tau ))\frac{1}{t\tau }\frac{1}{\pi }\int _{\mathcal {C}_{\tau ,t,j}}{\text {Im}}\varphi _{\tau ,t}(\xi )\,\mathrm {d}\xi \biggr \\ \le&\limsup _{\tau \nearrow t}\sum _{j\ne 1}w_{M,j}(\tau )w_{m,j}(\tau )\frac{1}{t\tau }\frac{1}{\pi }\int _{\mathcal {C}_{\tau ,t,j}}{\text {Im}}\varphi _{\tau ,t}(\xi )\,\mathrm {d}\xi \\ \le&\frac{4}{K}\left( \limsup _{\tau \nearrow t}\xi _{1,M}(\tau )^{1}+\limsup _{\tau \nearrow t}\xi _{1,m}(\tau )^{1}\right) \\&\times \limsup _{\tau \nearrow t}\sum _{j\ne 1}\frac{1}{t\tau }\frac{1}{\pi }\int _{\mathcal {C}_{\tau ,t,j}}{\text {Im}}\varphi _{\tau ,t}(\xi )\,\mathrm {d}\xi \\ \le&\frac{4L}{K}\bigl (\varphi _{t,1}(u_{M})U_{1}(t)\bigr +\varphi _{t,1}(u_{m})U_{1}(t)\bigr ), \end{aligned}$$
 (iv)We let \(a: =\lim _{k\rightarrow \infty }a_{k}\in \mathbb {C}\), and show thatexists, and is equal to a.^{9} To this end, let \((\tau _{m})_{m}\) be a sequence converging to t from below. We let$$\begin{aligned} \lim _{\tau \nearrow t}\frac{1}{\pi }\frac{1}{t\tau }\int _{\mathcal {C}_{\tau ,t,1}}{\text {Im}}\varphi _{\tau ,t}(\xi )\,\mathrm {d}\xi \end{aligned}$$With the help of Lemma 2.5 (3), we find that for all \(m,k\in \mathbb {N}\)$$\begin{aligned} b_{m}: =\frac{1}{\pi }\frac{1}{t\tau _{m}}\int _{\mathcal {C}_{\tau _{m},t,1}}{\text {Im}}\varphi _{\tau _{m},t}(\xi )\,\mathrm {d}\xi \in \left[ 0,1\right] . \end{aligned}$$Since \((b_{m})_{m}\) is a bounded sequence of real numbers, its limes superior exists. The estimate above yields, that for any \(k\in \mathbb {N}\)$$\begin{aligned} b_{m}a&\le \biggl \overbrace{b_{m}+\sum _{j\ne 1}w_{k,j}(\tau _{m})\frac{1}{\pi }\frac{1}{t\tau _{m}}\int _{\mathcal {C}_{\tau _{m},t,j}}{\text {Im}}\varphi _{\tau _{m},t}(\xi )\,\mathrm {d}\xi }^{\Delta _{k}(\tau _{m})}a\biggr \\&\quad +\biggl \sum _{j\ne 1}w_{k,j}(\tau _{m})\frac{1}{t\tau _{m}}\frac{1}{\pi }\int _{\mathcal {C}_{\tau _{m},t,j}}{\text {Im}}\varphi _{\tau _{m},t}(\xi )\,\mathrm {d}\xi \biggr \\&\le \Delta _{k}(\tau _{m})a+\frac{4}{K\xi _{1,k}(\tau _{m})}\sum _{j\ne 1}\frac{1}{t\tau _{m}}\frac{1}{\pi }\int _{\mathcal {C}_{\tau _{m},t,j}}{\text {Im}}\varphi _{\tau _{m},t}(\xi )\,\mathrm {d}\xi \\&\le \Delta _{k}(\tau _{m})a_{k}+a_{k}a+\frac{4L}{K}\frac{1}{\xi _{1,k}(\tau _{m})}. \end{aligned}$$Taking \(k\rightarrow \infty \), we get \(\limsup _{m\rightarrow \infty }b_{m}=a\). By arguing in the same way as above, we conclude that \(\liminf _{m\rightarrow \infty }b_{m}=a\), which entails \(\lim _{m\rightarrow \infty }b_{m}=a\). Hence, indeed$$\begin{aligned} \Bigl \limsup _{m\rightarrow \infty }b_{m}a\Bigr &\le \limsup _{m\rightarrow \infty }\Delta _{k}(\tau _{m})a_{k}+a_{k}a+\frac{4L}{K}\limsup _{m\rightarrow \infty }\xi _{1,k}(\tau _{m})^{1}\\&=a_{k}a_{k}+a_{k}a+\frac{4L}{K}\varphi _{t,1}(u_{k})U_{1}(t). \end{aligned}$$Lemma 3.4, and Corollary 2.7 yield that$$\begin{aligned} \lim _{\tau \nearrow t}\frac{1}{t\tau }\frac{1}{\pi }\int _{\mathcal {C}_{\tau ,t,j}}{\text {Im}}\varphi _{\tau ,t}(\xi )\,\mathrm {d}\xi =a. \end{aligned}$$By arguing as we did in the Steps (ii) to (iv), simply replacing \(\tau \) by T, and \(\tau \nearrow t\) by \(T\searrow t\), we find that$$\begin{aligned} a_{k}=\lim _{\tau \nearrow t}\Delta _{k}(\tau )=\lim _{\tau \nearrow t}D_{k}(t,\tau )=\dot{\varphi }_{t,1}(u_{k})=\lim _{T\searrow t}D_{k}(T,t). \end{aligned}$$$$\begin{aligned} \lim _{T\searrow t}\frac{1}{Tt}\frac{1}{\pi }\int _{\mathcal {C}_{t,T,j}}{\text {Im}}\varphi _{t,T}(\xi )\,\mathrm {d}\xi =a=\lim _{\tau \nearrow t}\frac{1}{t\tau }\frac{1}{\pi }\int _{\mathcal {C}_{\tau ,t,j}}{\text {Im}}\varphi _{\tau ,t}(\xi )\,\mathrm {d}\xi . \end{aligned}$$
5 The Multislit Equation
Now, we combine the results from the previous sections, and deduce a generalised Loewner equation.
Lemma 5.1
The map \(z\mapsto \partial _{t}\varphi _{t,s}(z)\) is defined for almost every \(t\in \left[ 0,s\right] \), and is holomorphic in \(\mathbb {H}\).
Proof
We are now able to prove our main result.
Proof of Theorem 1.4
 (i)Let \(0\le \tau \le t\le s\le 1.\) By Corollary 2.7, we already know that the function \(t\mapsto \varphi _{t,s}(z)\) is differentiable for almost every t. Therefore, it suffices to calculate its left derivative. We let \(\Delta _{j}\left( z,\tau ,t\right) : =\Xi _{j}\lambda _{j}\left( \tau ,t\right) \) whereand \(\Xi _{j}=\Xi _{j}(z,\tau ,t)\) is from (3.1) (see also Lemma 2.6). By Lemma 3.4,$$\begin{aligned} \lambda _{j}\left( \tau ,t\right) : =\frac{1}{t\tau }\frac{1}{\pi }\int _{\mathcal {C}_{\tau ,t,j}}{\text {Im}}\varphi _{\tau ,t}(\xi )\,\mathrm {d}\xi , \end{aligned}$$and consequently$$\begin{aligned} \Xi _{j}\rightarrow \frac{1}{\varphi _{t,s}(z)U_{j}(t)}\quad \text{ as } \quad \tau \nearrow t \end{aligned}$$Taking the limit \(\tau {\nearrow } t\) in (5.1), and interchanging limit with summation, by the dominated convergence theorem, yields that for almost every \(t\in [0,s]\), and \(z\in \mathbb {H}\) the equations$$\begin{aligned} \frac{\varphi _{t,s}(z)\varphi _{\tau ,s}(z)}{t\tau }=\sum _{j}\Delta _{j}(z,\tau ,t). \end{aligned}$$(5.1)hold true. This implies (1.2).$$\begin{aligned} \partial _{t}\varphi _{t,s}(z)=\sum _{j}\frac{\lambda _{j}(t)}{\varphi _{t,s}(z)U_{j}(t)}, \quad \mathrm {and} \quad \varphi _{0,s}(z)=h_{s}(z) \end{aligned}$$
 (ii)We now prove that \(\lambda _{j}\) actually has the properties it is claimed to have in Theorem 1.4. Using Lemma 5.1, we note that \(t\mapsto \partial _{t}\varphi _{t,1}(\varphi _{t,1}^{1}(U_{j}(t)+k^{1}))\) is measurable for k sufficiently large. We use Eq. (4.2) with the sequence \((k^{1})_{k}\), and writeHence, \(\lambda _{j}\) is measurable. Equation (4.1), in combination with Lemma 2.5 (3), yields that \(0\le \lambda _{j}(t)\le L\) holds almost everywhere, where L denotes a Lipschitz constant of \(t\mapsto {{\mathrm{hcap}}}(\Gamma _{t})\). Comparing coefficients in the expansion of$$\begin{aligned} \lambda _{j}(t)=\lim _{k\rightarrow \infty }\partial _{t}\varphi _{t,1}\bigl (\varphi _{t,1}^{1}(U_{j}(t)+k^{1})\bigr )\cdot k^{1}. \end{aligned}$$and the expansion of$$\begin{aligned} \sum _{k}\frac{\lambda _{k}(t)}{\varphi _{t,s}(z)U_{k}(t)}&=\frac{1}{\varphi _{t,s}(z)}\sum _{k}\sum _{m\ge 0}\lambda _{k}(t)\left( \frac{U_{k}(t)}{\varphi _{t,s}(z)}\right) ^{m}\\&=\frac{1}{\varphi _{t,s}(z)}\sum _{m\ge 0}\sum _{k}\lambda _{k}(t)\left( \frac{U_{k}(t)}{\varphi _{t,s}(z)}\right) ^{m},\quad \left z\right \rightarrow \infty , \end{aligned}$$yields (3). This concludes the proof.$$\begin{aligned} \frac{\partial }{\partial t}\varphi _{t,s}(z)&=\frac{\partial }{\partial t}\left( z+\frac{{{\mathrm{hcap}}}\Gamma _{t}{{\mathrm{hcap}}}\Gamma _{s}}{z}+O\bigl (\left z\right ^{2}\bigr )\right) \\&=\frac{1}{z}\frac{\partial }{\partial t}{{\mathrm{hcap}}}\Gamma _{t}+\frac{\partial }{\partial t}O\bigl (\left z\right ^{2}\bigr ),\quad \left z\right \rightarrow \infty , \end{aligned}$$
Moreover, we can now deduce a relation between the existence of the limits defining the weight functions in (4.1) at time \(t\in [0,1]\) and the validity of the Loewener equation (1.2) at t.
Theorem 5.2
 (1)
\(\tau \mapsto g_{\tau }(z)\) is differentiable at t for all \(z\in \mathbb {H}{\setminus }\Gamma \).
 (2)The following two limits exist, and are equal for any j$$\begin{aligned} \lim _{\tau \nearrow t}\frac{1}{t\tau }\frac{1}{\pi }\int _{\mathcal {C}_{\tau ,t,j}} {\text {Im}}\varphi _{\tau ,t}(\xi )\,\mathrm {d}\xi , \quad \lim _{\tau \searrow t}\frac{1}{\tau t}\frac{1}{\pi } \int _{\mathcal {C}_{t,\tau ,j}}{\text {Im}}\varphi _{t,\tau }(\xi )\,\mathrm {d}\xi . \end{aligned}$$
Proof
Theorem 4.1 yields that (1) implies (2). By carefully reviewing the proof of Theorem 1.4, we see that we could have written \(\tau \searrow t\) instead of \(\tau \nearrow t\). Hence, we conclude that (2) implies (1).\(\square \)
Remark 5.3
Footnotes
 1.
We do not specify the range of j, since it is irrelevant for our approach whether the multislit is an n slit or not.
 2.
In [11, p. 1], Loewner’s exact words were: “Das charakteristische Merkmal der angewandten Untersuchungsmethode besteht in der Ausnützung des Umstandes, daß bei Zusammensetzung von schlichten konformen Abbildungen wieder eine schlichte Abbildung entsteht, daß also die schlichten Abbildungen eine Gruppe bilden.” In English (translated by the authors): the characteristic property of the method applied here is the exploitation of the fact that the composition of two schlicht functions is, again, a schlicht function, that is, the schlicht mappings form a group.
 3.
The distinction between “left” and “right” is obtained in the following manner: Extend the simple curve \(\mathcal {J}_{t,T,k}\) to \(\infty \), thereby cutting \(\hat{\mathbb {H}}{\setminus }\mathcal {J}_{t,T}\) into two disjoint pathconnected components, the “left” component being the one, whose boundary contains an interval \((\infty ,b]\), where \(b>0\) is some real number.
 4.
Of course, \(\xi _{j},\xi _{j}'\) depend on z, t, T. However, for the ease of notation, we drop these dependencies from the notation. The only exception to this will be the proof of Theorem 1.4.
 5.
We say that a sequence \(\left( M_{k}\right) \) of sets tends to the point p if \(\sup _{m\in M_{k}}\left mp\right \) converges to 0.
 6.
By the kernel theorem, we refer to the following statement. Let \(\Omega _{n}\subset \mathbb {C}\) denote a sequence of domains. Let X and \(\Omega \) denote domains, where X is unbounded. If \(\Omega _n\) converges in the kernel sense to a domain \(\Omega \), and the sequence \(f_n:X\rightarrow \Omega _n\) of biholomorphisms satisfying (1.1) is locally bounded, then it converges locally uniformly to the unique biholomorphism \(f:X\rightarrow \Omega \) with (1.1). This can be proved in the same manner as the ordinary kernel convergence theorem.
 7.
A sequence of sets \((M_{k})\) shrinks to a point p if \(\bigcap _{k=1}^{\infty }M_{k}=\left\{ p\right\} \), and \(M_{k+1}\subseteq M_{k}\).
 8.
Note that \(U_{j}(t)\ne U_{k}(t)\) for \(j\ne k\) and \(U_{k}(t)\in \mathbb {R}\) which implies \( U_{j}(t) \in \mathbb {C}{\setminus }( \mathcal {\mathfrak {\overline{J}}}_{k,t,T} \cup \mathcal {\mathfrak {\overline{J}}}_{k,t,T}^{*} ) \).
 9.
Note that thus the limit a is in fact independent of the choice of the sequence \((u_{k})_{k}\).
Notes
Acknowledgements
Open access funding provided by Graz University of Technology. The authors would like to thank the anonymous referee for many valuable comments and a careful reading of an earlier version of this manuscript. We are especially grateful for the bringing to our attention of the references [1, 7]. The second author gratefully acknowledges the encouragement of Prof. Dr. Robert Tichy. Both the authors would also like to express their gratitude to Prof. Dr. Jörn Steuding for his advice.
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